From Last Time

1
From Last Time

• Hydrogen atom quantum numbers
• Quantum jumps, tunneling and measurements

Today

• Superposition of wave functions
• Indistinguishability
• Electron spin a new quantum effect
• The Hydrogen atom and the periodic table

2
Hydrogen Quantum Numbers

• Quantum numbers, n, l, ml
• n how charge is distributed radially around the
  nucleus. Average radial distance.
• This determines the energy
• l how spherical the charge distribution
• l 0, spherical, l 1 less spherical
• ml rotation of the charge around the z axis
• Rotation clockwise or
  counterclockwise and
  how fast
• Small energy
  differences for
  
  l and ml states

3
Measuring which slit

• Suppose we measure which slit the
  particle goes through?
• Interference pattern is destroyed!
• Wavefunction changes instantaneously over entire
  screen when measurement is made.
• Before superposition of wavefunctions through
  both slits. After only through one slit.

4
A superposition state

• Margarita or Beer?
• This QM state has equal superposition of two.
• Each outcome (drinking margarita, drinking beer)
  is equally likely.
• Actual outcome not determined until measurement
  is made (drink is tasted).

5
What is object before the measurement?

• What is this new drink?
• Is it really a physical object?
• Exactly how does the transformation from this
  object to a beer or a margarita take place?
• This is the collapse of the wavefunction.
• Details, probabilities in the collapse, depend on
  the wavefunction, and sometimes the measurement

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Not universally accepted

• Historically, not everyone agreed with this
  interpretation.
• Einstein was a notable opponent
• God does not play dice
• These ideas hotly debated in the early part of
  the 20th century.
• However, one more set of crazy ideas needed to
  understand the hydrogen atom and the periodic
  table.

7
Spin An intrinsic property

• Free electron, by itself in space, not only has a
  charge, but also acts like a bar magnet with a N
  and S pole.
• Since electron has charge, could explain this if
  the electron is spinning.
• Then resulting current loops would produce
  magnetic field just like a bar magnet.
• But as far as we can tell the electron is not
  spinning

8
Electron magnetic moment

• Why does it have a magnetic moment?
• It is a property of the electron in the same way
  that charge is a property.
• But there are some differences.
• Magnetic moment is a vector has a size and a
  direction
• Its size is intrinsic to the electron
• but the direction is variable.
• The bar magnet can point in different
  directions.

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Quantization of the direction

• But like everything in quantum mechanics, this
  magnitude and direction are quantized.
• And also like other things in quantum mechanics,
  if magnetic moment is very large, the
  quantization is not noticeable.
• But for an electron, the moment is very small.
• The quantization effect is very large.
• In fact, there is only one magnitude and two
  possible directions that the bar magnet can
  point.
• We call these spin up and spin down.
• Another quantum number spin up 1/2, down -1/2

10
Electron spin orientations
These are two different quantum states in which
an electron can exist.
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Other particles

• Other particles also have spin
• The proton is also a spin 1/2 particle.
• The neutron is a spin 1/2 particle.
• The photon is a spin 1 particle.
• The graviton is a spin 2 particle.

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Particle in a box

• We labeled the quantum states with an integer
• The lowest energy state was labeled n1
• This labeled the spatial properties of the
  wavefunction (wavelength, etc)
• Now we have an additional quantum property, spin.
  
• Spin quantum number could be 1/2 or -1/2

There are two quantum states with n1 Can write
them as
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Spin 1/2 particle in a box

• We talked about two quantum states
• In isolated space, which has lower energy?

A. B. C. Both same
An example of degeneracy two quantum states that
have exactly the same energy.
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Indistinguishability

• Another property of quantum particles
• All electrons are ABSOLUTELY identical.
• Never true at the macroscopic scale.
• On the macroscopic scale, there is always some
  aspect that distinguishes two objects.
• Perhaps color, or rough or smooth surface
• Maybe a small scratch somewhere.
• Experimentally, no one has ever found any
  differences between electrons.

15
Indinstinguishability and QM

• Quantum Mechanics says that electrons are
  absolutely indistinguishable.Treats this as an
  experimental fact.
• For instance, it is impossible to follow an
  electron throughout its orbit in order to
  identify it later.
• We can still label the particles, for instance
• Electron 1, electron 2, electron 3
• But the results will be meaningful only if we
  preserve indistinguishability.
• Find that this leads to some unusual consiquenses

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Example 2 electrons on an atom

• Probability of finding an electron at a location
  is given by the square of the wavefunction.
• We have two electrons, so the question we would
  is ask is
• How likely is it to find one electron at location
  r1 and the other electron at r2?

17

• Suppose we want to describe the state with

one electron in a 3s state
and one electron in a 3d state
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On the atom, they look like this. (Both on the
same atom).

• Must describe this with a wavefunction that says
• We have two electrons
• One of the electrons is in s-state, one in
  d-state
• Also must preserve indistinguishability

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Question

• Which one of these states doesnt change when
  we switch particle labels.

A.
B.
C.
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Preserves indistinguishability
Wavefunction unchanged
21
Physically measurable quantities

• How can we label particles, but still not
  distinguish them?
• What is really meant is that no physically
  measurable results can depend on how we label the
  particles.
• One physically measurable result is the
  probability of finding an electron in a
  particular spatial location.

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Probabilities

• The probability of finding the particles at
  particular locations is the square of the
  wavefunction.
• Indistinguishability says that these
  probabilities cannot change if we switch the
  labels on the particles.
• However the wavefunction could change, since it
  is not directly measurable.(Probability is the
  square of the wavefunction)

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Two possible wavefunctions

• Two possible symmetries of the wavefunction,
  that keep the probability unchanged when we
  exchange particle labels
• The wavefunction does not change Symmetric
• The wavefunction changes sign only
  Antisymmetric

In both cases the square is unchanged
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Another possible wavefunction

Electron 1 in s-state
Electron 2 in d-state
Wavefunction changes sign
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Spin-statistics theorem

• In both cases the probability is preserved, since
  it is the square of the wavefunction.
• Can be shown that
• Integer spin particles (e.g. photons) have
  wavefunctions with sign (symmetric)These
  types of particles are called Bosons
• Half-integer spin particles (e.g. electrons)have
  wavefunctions with - sign (antisymmetric)These
  types of particles are called Fermions

26
So what?

• Fermions - antisymmetric wavefunction

Try to put two Fermions in the same quantum state
(for instance both in the s-state)
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Pauli exclusion principle

• Only wave function permitted by
  indistinghishability is exactly zero. This means
  that this never happens.
• Cannot put two Fermions in same quantum state
• This came entirely from indisinguishability,
  that electrons are identical.
• Without this,
• there elements would not have diff. chem. props.,
• properties of metals would be different,
• neutron stars would collapse.

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Include spin

• We labeled the states by their quantum numbers.
  One quantum number for each spatial dimension.
• Now there is an extra quantum number spin.
• A quantum state is specified by its space part
  and also its spin part.
• An atom with several electrons filling quantum
  states starting with the lowest energy, filling
  quantum states until electrons are used.

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Putting electrons on atom

• Electrons are Fermions
• Only one electron per quantum state

unoccupied
occupied
n1 states
Hydrogen 1 electron one quantum states occupied
Helium 2 electronstwo quantum states occupied
n1 states
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Other elements

• More electrons requires next higher energy states
• Lithium three electrons

Other states empty
Elements with more electrons have more complex
states occupied
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